Articles of functional analysis

Contradiction achieved with the Pettis Measurability Theorem?

$\bf{\text{(Pettis Measurability Theorem)}}$ Let $(\Omega,\Sigma,\mu)$ be a $\sigma$-finite measure. The following are equivalent for $f:\Omega\to X$. (i) $f$ is $\mu$-measurable. (ii) $f$ is weakly $\mu$-measurable and $\mu$-essentially separately valued. (iii) $f$ is Borel measurable and $\mu$-essentially separately valued. $\bf{\text{Relevant Definitions:}}$ A function $f:\Omega\to X$ is simple if it assumes only finitely many values. That is, […]

Show the Volterra Operator is compact using only the definition of compact

The Volterra operator $V:L^{2}[0,1]\rightarrow L^{2}[0,1]$ is defined by $(Vf)(x)=\int_0^xf(t)dt$. I am wondering if it can be shown that $V$ is compact by definition – that is, either that $V$ maps bounded sets to precompact sets, or equivalently, that for any bounded sequence $(f_n)$ in the domain, $\{Vf_n\}$ has a convergent subsequence. I have seen an […]

Norm of the Resolvent

Let $\mathbb{H}$ be a Hilbert space, $A$ a self-adjoint operator with domain $D_{A}$, $R_{A}$ the resolvent of $A$, and $z$ a point in the resolvent set $\rho(A)$. How could you prove the inequality \begin{equation} ||R_{A}(z)|| \leq 1/ d(z,\sigma(A)), \end{equation} where $\sigma(A)$ is the spectrum of $A$, and $d(z,\sigma(A))$ the distance of $z$ from $\sigma(A)$? I […]

Duality of $L^p$ and $L^q$

If $X$ is an arbitrary measure space, I already know with proof that $L^p(X)$ and $L^q(X)$ are mutually duals as Banach spaces, when $1<p$ and $p$, $q$ are dual indices. I also know a different proof that only works when $X$ is sigma finite, but then it establishes also that the dual of $L^1(X)$ is […]

Range conditions on a linear operator

While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this: I have a linear operator $A:L^2(\Bbb{R}^3)\rightarrow L^2(\Bbb{R}^4)$, that is a mapping which takes functions of three variables to functions of four variables. Then, “because the range function depends on 4 […]

About Cauchy sequence

Suppose $(X,d)$ be a metric space. Let $(a_n)$ be a sequence in $X$ such that $(a_{2n-1})$ and $(a_{2n})$ has no Cauchy subsequence. Is it also true that $(a_n)$ has no Cauchy subsequence? Let $A=\{a_{2n}:n\in\mathbb{N}\}$, is it true every Cauchy sequence in $A$ is constant?

If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism?

$X$ and $Y$ denote Hilbert spaces. If $T:X \to Y$ is a linear homeomorphism, is its adjoint $T^*$ a linear homeomorphism? Homeomorphism means continuous map with continuous inverse. I think the answer is yes, the only thing I am unable to show is what is the inverse of $T^*$ and if it is continuous?

Is the composition function again in $L^2$

Let $f \in L^2[a,b]$. 1- In what condition(s) on a function $g:[a,b]\rightarrow [a,b]$ we can get $$f \circ g \in L^2[a,b]?$$ 2- In what condition(s) on $g:[a,b]\rightarrow [a,b]$, the operator $T:L^2[a,b] \rightarrow L^2[a,b]$ which is defined by $Tf = f \circ g$, is a bounded linear operator?

Spectrum of a Self-Adjoint Operator is Real

Preparing for an exam in functional analysis, I’m trying to show that for a self-adjoint operator $A$, $\sigma(A) \subset \mathbb{R}$. I came across the following proof in the book (or rather, lecture notes) we’re using for the course. The proof is even stronger, giving bounds for the spectrum. However, I have issue with the proof. […]

Does every Banach space admit a continuous injection to a non-closed subspace of another Banach space?

Let $(V, \lVert\,\rVert)$ be a Banach space. I want to produce a non-complete norm $\lVert\,\rVert’$ on it such that $\lVert v\rVert’ \leq \lVert v\rVert$ for all $v$ in $V$. Given a continuous injection $\varphi\colon V \to W$ with a non-closed image, taking $\lVert v\rVert’ = C \lVert \varphi(v)\rVert$ for some $C$ provided by continuity (preimage […]